OpenAI announced on August 1, 2026, that an internal preview version of its upcoming model Astra has solved ten long-standing problems in mathematics and theoretical computer science. Each proof is available as a machine-checkable Lean certificate on GitHub. According to the company, the computational power for all ten solutions cost around $2,000.
Proofs range from group theory to cryptography
According to the OpenAI blog post, the ten results span six subfields: high-dimensional geometry, coding theory, group theory, quantum complexity, lattice theory, and extremal combinatorics. Astra has constructed an explicit non-sofic group for the first time—a question in group theory that has been open since 1999—and disproved Connes’ rigidity conjecture about von Neumann algebras. Additionally, the model proved the long-standing Ehrhart volume conjecture from convex geometry and improved the general upper density bound of sphere packing for the first time since 1978.
Furthermore, there is a parallel repetition theorem for two-player quantum games, a new lower bound for the circuit complexity of the permanent, and advancements on the closest vector problem, which is relevant for the transition to post-quantum encryption. Astra also improved known bounds in coding theory for binary and spherical codes. Three additional solutions pertain to Erdős problems, including the multicolored Ramsey number from Problem 183. Accompanying this, OpenAI published a 249-page manuscript with the complete argumentation; the scientific community had made no progress on most of the ten questions for at least ten years.
Lean certificates replace traditional peer review
Lean 4 is a software tool that formally verifies mathematical arguments step by step, rather than submitting them solely to human reviewers. Each of the ten proofs is formalized within it and is available along with a human-written manuscript and the model’s argument chain in the GitHub repository. These machine-checkable certificates are intended to allow for independent verification even before journals review the results. Traditional peer review of mathematical articles often takes many months because reviewers must manually verify each proof step individually.
OpenAI researcher Noam Brown writes on X that the results are an important step for scientific reasoning. He also acknowledges that Astra has so far failed on the Millennium Prize Problems and that the test run per task used relatively little computational power—more computing time could therefore yield further results. Human experts translated the model outputs into the published manuscripts; a complete external review by the mathematics community is still pending, as is an assessment of how much the proofs build on already known subarguments.
Astra undergoes voluntary US model review
According to OpenAI, Astra is a system of multiple collaborating agents that is supposed to work on a problem over hours or days, rather than just answering individual queries—mathematics serves as a showcase for this capability. OpenAI CEO Sam Altman has already shown the model to lawmakers and government representatives in Washington. Before a broader release, Astra is to undergo the voluntary review framework established by the US government via an executive order on June 2, 2026, for particularly powerful AI models: developers can make a model accessible to the government up to 30 days before its release.
Just in July, OpenAI took another internal model offline after repeated breakouts from its sandbox—that system had also previously disproved a mathematical conjecture. The model GPT-5.6 Sol Ultra had presented a purported proof for a fifty-year-old problem in July, which experts were still reviewing at that time. The frequency of such cases shows how much mathematics currently serves as a testing ground for the capabilities of upcoming models.
It remains to be seen whether independent mathematicians can confirm all ten certificates before Astra is tested beyond curated proof showcases. OpenAI has not yet announced a release date or a specific product name—whether as GPT-6 or as a separate tier alongside Sol, Terra, and Luna. The real crux will be whether the demonstrated problem-solving ability can also be confirmed outside of selected math problems in real research and engineering tasks.
